A high-tier character is a character that for some reason or other, is a cut above most of the rest of the cast in the game.
Usually, high tier just means that a character is more viable for winning, but has nothing extremely special to them. However, in some games, some (perfectly normal-seeming) characters are unfairly better than others, breaking the game balance.
There is no such thing as a perfectly-balanced game, but there always should be some semblance of balance for a game to be any good. Still, there always will be some characters that are good choices in many matchups, giving you the best chance of winning when it comes to character selection alone. In particular, developers cannot balance a very large cast all that well, so games with lots of characters (like the whole KOF series) will make it apparent who the top-tier characters are, because practically everyone is using them.
Going beyond characters who are merely good, there are those that are far and away better than the others (often to bugs and other oversights). Just using them can get yourself shunned (though this phenomenon is usually restricted to poorly-balanced games). This includes Toki in Hokuto no Ken, Foxy in KOF 2001, Duo Long in KOF 2003, Gim Gap-hwan in Fatal Fury Special, and Tokinomiya Kamui in Arcana Heart, among others.
In home ports, there tend to be secret characters and bosses that can only be used in the ports; these tend not to be well-tested (or not intended to be fair, for that matter), so they are often only good for money matches and friendly company.
On occasion, there are cases where characters initially believed to be low-tier turn out to be much more effective after people research the game in more depth.
Also, sequels that keep the same characters around tend to tweak the balance a bit, so a character does not often keep their rank within the pecking order unless the developers do it intentionally (e.g., to keep heroes popular, and to keep joke characters a joke).
Characters high in a tier list usually have very basic advantages, such as moves with quick startup, good priority/positioning to their attack hitboxes, fewer openings, and/or high offensive output/throughput.
Most characters will have a couple of these respective attributes, making plenty of them lopsided in some way as a matter of course; but among them, there will be some that may be advantaged due to the game’s systems, etc., so that their weak spots aren’t necessarily as exploitable compared to others. These properties often determine who the high-tier characters are soon after a game’s release.